Pushforward measure

In mathematics, a pushforward measure (also push forward or push-forward) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.

Definition

Given measurable spaces ("X"₁, Σ₁) and ("X"₂, Σ₂), a measurable function "f" : "X"₁ → "X"₂ and a measure "μ" : Σ₁ → [0, +∞] , the pushforward of "μ" is defined to be the measure "f"_∗("μ") : Σ₂ → [0, +∞] given by

:$(f\_\{*\}\; (mu))\; (B)\; =\; mu\; left(\; f^\{-1\}\; (B)\; ight)\; mbox\{\; for\; \}\; B\; in\; Sigma\_\{2\}.$

This definition applies "mutatis mutandis" for a signed or complex measure.

Examples and applications

* A natural "Lebesgue measure" on the unit circle S¹ (here thought of as a subset of the complex plane C) may be defined using a push-forward construction and Lebesgue measure "λ" on the real line R. Let "λ" also denote the restriction of Lebesgue measure to the interval [0, 2"π") and let "f" : [0, 2"π") → S¹ be the natural bijection defined by "f"("t") = exp("i" "t"). The natural "Lebesgue measure" on S¹ is then the push-forward measure "f"_∗("λ"). The measure "f"_∗("λ") might also be called "arc length measure" or "angle measure", since the "f"_∗("λ")-measure of an arc in S¹ is precisely is its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)

* The previous example extends nicely to give a natural "Lebesgue measure" on the "n"-dimensional torus T^"n". The previous example is a special case, since S¹ = T¹. This Lebesgue measure on T^"n" is, up to normalization, the Haar measure for the compact, connected Lie group T^"n".

* Gaussian measures on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: a Borel measure "γ" on a separable Banach space "X" is called Gaussian if the push-forward of "γ" by any non-zero linear functional in the continuous dual space to "X" is a Gaussian measure on R.

* Consider a measurable function "f" : "X" → "X" and the composition of "f" with itself "n" times:

::$f^\{(n)\}\; =\; underbrace\{f\; circ\; f\; circ\; dots\; circ\; f\}\_\{n\; mathrm\{,\; times\; :\; X\; o\; X.$

: This forms a measurable dynamical system. It is often of interest in the study of such systems to find a measure "μ" on "X" that the map "f" leaves unchanged, a so-called invariant measure, one for which "f"_∗("μ") = "μ".

* One can also consider quasi-invariant measures for such a dynamical system: a measure "μ" on "X" is called quasi-invariant under "f" if the push-forward of "μ" by "f" is merely equivalent to the original measure "μ", not necessarily equal to it.